% TIMEFORMAT='%3R'; { time (exec 2>&1; /home/martin/bin/satallax -E /home/martin/.isabelle/contrib/e-2.5-1/x86_64-linux/eprover -p tstp -t 5 /home/martin/judgement-day/tptp-thf/tptp/Fundamental_Theorem_Algebra/prob_1115__5378410_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:35:58.312

% Could-be-implicit typings (3)
thf(ty_n_t__Polynomial__Opoly_Itf__a_J, type,
    poly_a : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_tf__a, type,
    a : $tType).

% Explicit typings (20)
thf(sy_c_Fundamental__Theorem__Algebra__Mirabelle__sywschxjbb_Opsize_001tf__a, type,
    fundam247907092size_a : poly_a > nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Polynomial__Opoly_Itf__a_J, type,
    one_one_poly_a : poly_a).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Polynomial__Opoly_Itf__a_J, type,
    times_times_poly_a : poly_a > poly_a > poly_a).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Polynomial__Opoly_Itf__a_J, type,
    zero_zero_poly_a : poly_a).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Polynomial__Opoly_Itf__a_J, type,
    semiri1217733995poly_a : nat > poly_a).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Polynomial_Odegree_001tf__a, type,
    degree_a : poly_a > nat).
thf(sy_c_Polynomial_Ois__zero_001tf__a, type,
    is_zero_a : poly_a > $o).
thf(sy_c_Polynomial_Opcompose_001tf__a, type,
    pcompose_a : poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Opoly__cutoff_001tf__a, type,
    poly_cutoff_a : nat > poly_a > poly_a).
thf(sy_c_Polynomial_Opseudo__mod_001tf__a, type,
    pseudo_mod_a : poly_a > poly_a > poly_a).
thf(sy_c_Polynomial_Osynthetic__div_001tf__a, type,
    synthetic_div_a : poly_a > a > poly_a).
thf(sy_v_p, type,
    p : poly_a).
thf(sy_v_q, type,
    q : poly_a).

% Relevant facts (143)
thf(fact_0__092_060open_062q_A_061_A0_092_060close_062, axiom,
    ((q = zero_zero_poly_a))). % \<open>q = 0\<close>
thf(fact_1_lq, axiom,
    ((~ ((p = zero_zero_poly_a))))). % lq
thf(fact_2_mult__zero__left, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ zero_zero_poly_a @ A) = zero_zero_poly_a)))). % mult_zero_left
thf(fact_3_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_4_mult__zero__right, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ A @ zero_zero_poly_a) = zero_zero_poly_a)))). % mult_zero_right
thf(fact_5_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_6_mult__eq__0__iff, axiom,
    ((![A : poly_a, B : poly_a]: (((times_times_poly_a @ A @ B) = zero_zero_poly_a) = (((A = zero_zero_poly_a)) | ((B = zero_zero_poly_a))))))). % mult_eq_0_iff
thf(fact_7_mult__eq__0__iff, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) = (((A = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_8_mult__cancel__left, axiom,
    ((![C : poly_a, A : poly_a, B : poly_a]: (((times_times_poly_a @ C @ A) = (times_times_poly_a @ C @ B)) = (((C = zero_zero_poly_a)) | ((A = B))))))). % mult_cancel_left
thf(fact_9_mult__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_left
thf(fact_10_mult__cancel__right, axiom,
    ((![A : poly_a, C : poly_a, B : poly_a]: (((times_times_poly_a @ A @ C) = (times_times_poly_a @ B @ C)) = (((C = zero_zero_poly_a)) | ((A = B))))))). % mult_cancel_right
thf(fact_11_mult__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_right
thf(fact_12_lgpq, axiom,
    ((ord_less_nat @ (degree_a @ q) @ (degree_a @ p)))). % lgpq
thf(fact_13_mult__poly__0__left, axiom,
    ((![Q : poly_a]: ((times_times_poly_a @ zero_zero_poly_a @ Q) = zero_zero_poly_a)))). % mult_poly_0_left
thf(fact_14_mult__poly__0__right, axiom,
    ((![P : poly_a]: ((times_times_poly_a @ P @ zero_zero_poly_a) = zero_zero_poly_a)))). % mult_poly_0_right
thf(fact_15_mult__not__zero, axiom,
    ((![A : poly_a, B : poly_a]: ((~ (((times_times_poly_a @ A @ B) = zero_zero_poly_a))) => ((~ ((A = zero_zero_poly_a))) & (~ ((B = zero_zero_poly_a)))))))). % mult_not_zero
thf(fact_16_mult__not__zero, axiom,
    ((![A : nat, B : nat]: ((~ (((times_times_nat @ A @ B) = zero_zero_nat))) => ((~ ((A = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_17_divisors__zero, axiom,
    ((![A : poly_a, B : poly_a]: (((times_times_poly_a @ A @ B) = zero_zero_poly_a) => ((A = zero_zero_poly_a) | (B = zero_zero_poly_a)))))). % divisors_zero
thf(fact_18_divisors__zero, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) => ((A = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_19_no__zero__divisors, axiom,
    ((![A : poly_a, B : poly_a]: ((~ ((A = zero_zero_poly_a))) => ((~ ((B = zero_zero_poly_a))) => (~ (((times_times_poly_a @ A @ B) = zero_zero_poly_a)))))))). % no_zero_divisors
thf(fact_20_no__zero__divisors, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_21_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_22_degree__0, axiom,
    (((degree_a @ zero_zero_poly_a) = zero_zero_nat))). % degree_0
thf(fact_23_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_24_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_25_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_26_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_27_degree__mult__eq__0, axiom,
    ((![P : poly_a, Q : poly_a]: (((degree_a @ (times_times_poly_a @ P @ Q)) = zero_zero_nat) = (((P = zero_zero_poly_a)) | ((((Q = zero_zero_poly_a)) | ((((~ ((P = zero_zero_poly_a)))) & ((((~ ((Q = zero_zero_poly_a)))) & (((((degree_a @ P) = zero_zero_nat)) & (((degree_a @ Q) = zero_zero_nat))))))))))))))). % degree_mult_eq_0
thf(fact_28_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B))))))). % linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_29_mult__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C))))))). % mult_strict_right_mono
thf(fact_30_mult__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B))))))). % mult_strict_left_mono
thf(fact_31_zero__less__mult__pos2, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ B @ A)) => ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ B)))))). % zero_less_mult_pos2
thf(fact_32_zero__less__mult__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ A @ B)) => ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ B)))))). % zero_less_mult_pos
thf(fact_33_mult__pos__neg2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (times_times_nat @ B @ A) @ zero_zero_nat)))))). % mult_pos_neg2
thf(fact_34_mult__pos__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ zero_zero_nat @ (times_times_nat @ A @ B))))))). % mult_pos_pos
thf(fact_35_mult__pos__neg, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (times_times_nat @ A @ B) @ zero_zero_nat)))))). % mult_pos_neg
thf(fact_36_mult__neg__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ zero_zero_nat) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ (times_times_nat @ A @ B) @ zero_zero_nat)))))). % mult_neg_pos
thf(fact_37_zero__reorient, axiom,
    ((![X : poly_a]: ((zero_zero_poly_a = X) = (X = zero_zero_poly_a))))). % zero_reorient
thf(fact_38_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_39_mult_Oleft__commute, axiom,
    ((![B : poly_a, A : poly_a, C : poly_a]: ((times_times_poly_a @ B @ (times_times_poly_a @ A @ C)) = (times_times_poly_a @ A @ (times_times_poly_a @ B @ C)))))). % mult.left_commute
thf(fact_40_mult_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((times_times_nat @ B @ (times_times_nat @ A @ C)) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.left_commute
thf(fact_41_mult_Ocommute, axiom,
    ((times_times_poly_a = (^[A2 : poly_a]: (^[B2 : poly_a]: (times_times_poly_a @ B2 @ A2)))))). % mult.commute
thf(fact_42_mult_Ocommute, axiom,
    ((times_times_nat = (^[A2 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A2)))))). % mult.commute
thf(fact_43_mult_Oassoc, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ (times_times_poly_a @ A @ B) @ C) = (times_times_poly_a @ A @ (times_times_poly_a @ B @ C)))))). % mult.assoc
thf(fact_44_mult_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.assoc
thf(fact_45_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : poly_a, B : poly_a, C : poly_a]: ((times_times_poly_a @ (times_times_poly_a @ A @ B) @ C) = (times_times_poly_a @ A @ (times_times_poly_a @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_46_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_47_mult__right__cancel, axiom,
    ((![C : poly_a, A : poly_a, B : poly_a]: ((~ ((C = zero_zero_poly_a))) => (((times_times_poly_a @ A @ C) = (times_times_poly_a @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_48_mult__right__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (A = B)))))). % mult_right_cancel
thf(fact_49_mult__left__cancel, axiom,
    ((![C : poly_a, A : poly_a, B : poly_a]: ((~ ((C = zero_zero_poly_a))) => (((times_times_poly_a @ C @ A) = (times_times_poly_a @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_50_mult__left__cancel, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (A = B)))))). % mult_left_cancel
thf(fact_51_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_52_mult__less__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M @ N))))))). % mult_less_cancel2
thf(fact_53_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_54_nat__0__less__mult__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ M @ N)) = (((ord_less_nat @ zero_zero_nat @ M)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % nat_0_less_mult_iff
thf(fact_55_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_56_nat__mult__less__cancel__disj, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M @ N))))))). % nat_mult_less_cancel_disj
thf(fact_57_pseudo__mod_I2_J, axiom,
    ((![G : poly_a, F : poly_a]: ((~ ((G = zero_zero_poly_a))) => (((pseudo_mod_a @ F @ G) = zero_zero_poly_a) | (ord_less_nat @ (degree_a @ (pseudo_mod_a @ F @ G)) @ (degree_a @ G))))))). % pseudo_mod(2)
thf(fact_58_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_59_mult__is__0, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_60_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_61_mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_62_mult__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N @ K)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_63_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_64_nat__mult__eq__cancel__disj, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((K = zero_zero_nat)) | ((M = N))))))). % nat_mult_eq_cancel_disj
thf(fact_65_nat__mult__less__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K) => ((ord_less_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)) = (ord_less_nat @ M @ N)))))). % nat_mult_less_cancel1
thf(fact_66_nat__mult__eq__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K) => (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (M = N)))))). % nat_mult_eq_cancel1
thf(fact_67_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_68_linorder__neqE__nat, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % linorder_neqE_nat
thf(fact_69_infinite__descent0, axiom,
    ((![P2 : nat > $o, N : nat]: ((P2 @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P2 @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P2 @ M2)))))))) => (P2 @ N)))))). % infinite_descent0
thf(fact_70_infinite__descent, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P2 @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P2 @ M2))))))) => (P2 @ N))))). % infinite_descent
thf(fact_71_nat__less__induct, axiom,
    ((![P2 : nat > $o, N : nat]: ((![N2 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N2) => (P2 @ M2))) => (P2 @ N2))) => (P2 @ N))))). % nat_less_induct
thf(fact_72_mult__less__mono2, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ K @ I) @ (times_times_nat @ K @ J))))))). % mult_less_mono2
thf(fact_73_mult__less__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ I @ K) @ (times_times_nat @ J @ K))))))). % mult_less_mono1
thf(fact_74_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_75_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_76_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_77_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_78_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_79_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_80_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_81_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_82_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_83_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_84_psize__eq__0__iff, axiom,
    ((![P : poly_a]: (((fundam247907092size_a @ P) = zero_zero_nat) = (P = zero_zero_poly_a))))). % psize_eq_0_iff
thf(fact_85_synthetic__div__eq__0__iff, axiom,
    ((![P : poly_a, C : a]: (((synthetic_div_a @ P @ C) = zero_zero_poly_a) = ((degree_a @ P) = zero_zero_nat))))). % synthetic_div_eq_0_iff
thf(fact_86_is__zero__null, axiom,
    ((is_zero_a = (^[P3 : poly_a]: (P3 = zero_zero_poly_a))))). % is_zero_null
thf(fact_87_poly__cutoff__0, axiom,
    ((![N : nat]: ((poly_cutoff_a @ N @ zero_zero_poly_a) = zero_zero_poly_a)))). % poly_cutoff_0
thf(fact_88_pcompose__eq__0, axiom,
    ((![P : poly_a, Q : poly_a]: (((pcompose_a @ P @ Q) = zero_zero_poly_a) => ((ord_less_nat @ zero_zero_nat @ (degree_a @ Q)) => (P = zero_zero_poly_a)))))). % pcompose_eq_0
thf(fact_89_pcompose__0, axiom,
    ((![Q : poly_a]: ((pcompose_a @ zero_zero_poly_a @ Q) = zero_zero_poly_a)))). % pcompose_0
thf(fact_90_synthetic__div__0, axiom,
    ((![C : a]: ((synthetic_div_a @ zero_zero_poly_a @ C) = zero_zero_poly_a)))). % synthetic_div_0
thf(fact_91_pcompose__mult, axiom,
    ((![P : poly_a, Q : poly_a, R : poly_a]: ((pcompose_a @ (times_times_poly_a @ P @ Q) @ R) = (times_times_poly_a @ (pcompose_a @ P @ R) @ (pcompose_a @ Q @ R)))))). % pcompose_mult
thf(fact_92_degree__pcompose, axiom,
    ((![P : poly_a, Q : poly_a]: ((degree_a @ (pcompose_a @ P @ Q)) = (times_times_nat @ (degree_a @ P) @ (degree_a @ Q)))))). % degree_pcompose
thf(fact_93_poly__cutoff__1, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((poly_cutoff_a @ N @ one_one_poly_a) = zero_zero_poly_a)) & ((~ ((N = zero_zero_nat))) => ((poly_cutoff_a @ N @ one_one_poly_a) = one_one_poly_a)))))). % poly_cutoff_1
thf(fact_94_psize__def, axiom,
    ((fundam247907092size_a = (^[P3 : poly_a]: (if_nat @ (P3 = zero_zero_poly_a) @ zero_zero_nat @ (suc @ (degree_a @ P3))))))). % psize_def
thf(fact_95_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_96_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_97_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_98_mult_Oright__neutral, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ A @ one_one_poly_a) = A)))). % mult.right_neutral
thf(fact_99_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_100_mult_Oleft__neutral, axiom,
    ((![A : poly_a]: ((times_times_poly_a @ one_one_poly_a @ A) = A)))). % mult.left_neutral
thf(fact_101_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_102_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_103_Suc__mono, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (suc @ M) @ (suc @ N)))))). % Suc_mono
thf(fact_104_Suc__less__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) = (ord_less_nat @ M @ N))))). % Suc_less_eq
thf(fact_105_degree__of__nat, axiom,
    ((![N : nat]: ((degree_a @ (semiri1217733995poly_a @ N)) = zero_zero_nat)))). % degree_of_nat
thf(fact_106_mult__cancel__left1, axiom,
    ((![C : poly_a, B : poly_a]: ((C = (times_times_poly_a @ C @ B)) = (((C = zero_zero_poly_a)) | ((B = one_one_poly_a))))))). % mult_cancel_left1
thf(fact_107_mult__cancel__left2, axiom,
    ((![C : poly_a, A : poly_a]: (((times_times_poly_a @ C @ A) = C) = (((C = zero_zero_poly_a)) | ((A = one_one_poly_a))))))). % mult_cancel_left2
thf(fact_108_mult__cancel__right1, axiom,
    ((![C : poly_a, B : poly_a]: ((C = (times_times_poly_a @ B @ C)) = (((C = zero_zero_poly_a)) | ((B = one_one_poly_a))))))). % mult_cancel_right1
thf(fact_109_mult__cancel__right2, axiom,
    ((![A : poly_a, C : poly_a]: (((times_times_poly_a @ A @ C) = C) = (((C = zero_zero_poly_a)) | ((A = one_one_poly_a))))))). % mult_cancel_right2
thf(fact_110_of__nat__0, axiom,
    (((semiri1217733995poly_a @ zero_zero_nat) = zero_zero_poly_a))). % of_nat_0
thf(fact_111_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_112_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_nat = (semiri1382578993at_nat @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_113_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri1382578993at_nat @ M) = zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_114_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_115_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_116_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_117_of__nat__mult, axiom,
    ((![M : nat, N : nat]: ((semiri1217733995poly_a @ (times_times_nat @ M @ N)) = (times_times_poly_a @ (semiri1217733995poly_a @ M) @ (semiri1217733995poly_a @ N)))))). % of_nat_mult
thf(fact_118_of__nat__mult, axiom,
    ((![M : nat, N : nat]: ((semiri1382578993at_nat @ (times_times_nat @ M @ N)) = (times_times_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % of_nat_mult
thf(fact_119_mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = (suc @ zero_zero_nat)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % mult_eq_1_iff
thf(fact_120_one__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: (((suc @ zero_zero_nat) = (times_times_nat @ M @ N)) = (((M = (suc @ zero_zero_nat))) & ((N = (suc @ zero_zero_nat)))))))). % one_eq_mult_iff
thf(fact_121_degree__1, axiom,
    (((degree_a @ one_one_poly_a) = zero_zero_nat))). % degree_1
thf(fact_122_Nat_OlessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ I @ K) => ((~ ((K = (suc @ I)))) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K = (suc @ J2))))))))))))). % Nat.lessE
thf(fact_123_Suc__lessD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ N) => (ord_less_nat @ M @ N))))). % Suc_lessD
thf(fact_124_Suc__lessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ (suc @ I) @ K) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K = (suc @ J2)))))))))))). % Suc_lessE
thf(fact_125_Suc__lessI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => ((~ (((suc @ M) = N))) => (ord_less_nat @ (suc @ M) @ N)))))). % Suc_lessI
thf(fact_126_less__SucE, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) => ((~ ((ord_less_nat @ M @ N))) => (M = N)))))). % less_SucE
thf(fact_127_less__SucI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ M @ (suc @ N)))))). % less_SucI
thf(fact_128_Ex__less__Suc, axiom,
    ((![N : nat, P2 : nat > $o]: ((?[I2 : nat]: (((ord_less_nat @ I2 @ (suc @ N))) & ((P2 @ I2)))) = (((P2 @ N)) | ((?[I2 : nat]: (((ord_less_nat @ I2 @ N)) & ((P2 @ I2)))))))))). % Ex_less_Suc
thf(fact_129_less__Suc__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) = (((ord_less_nat @ M @ N)) | ((M = N))))))). % less_Suc_eq
thf(fact_130_not__less__eq, axiom,
    ((![M : nat, N : nat]: ((~ ((ord_less_nat @ M @ N))) = (ord_less_nat @ N @ (suc @ M)))))). % not_less_eq
thf(fact_131_All__less__Suc, axiom,
    ((![N : nat, P2 : nat > $o]: ((![I2 : nat]: (((ord_less_nat @ I2 @ (suc @ N))) => ((P2 @ I2)))) = (((P2 @ N)) & ((![I2 : nat]: (((ord_less_nat @ I2 @ N)) => ((P2 @ I2)))))))))). % All_less_Suc
thf(fact_132_Suc__less__eq2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ (suc @ N) @ M) = (?[M3 : nat]: (((M = (suc @ M3))) & ((ord_less_nat @ N @ M3)))))))). % Suc_less_eq2
thf(fact_133_less__antisym, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) => (M = N)))))). % less_antisym
thf(fact_134_Suc__less__SucD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) => (ord_less_nat @ M @ N))))). % Suc_less_SucD
thf(fact_135_less__trans__Suc, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ J @ K) => (ord_less_nat @ (suc @ I) @ K)))))). % less_trans_Suc
thf(fact_136_less__Suc__induct, axiom,
    ((![I : nat, J : nat, P2 : nat > nat > $o]: ((ord_less_nat @ I @ J) => ((![I3 : nat]: (P2 @ I3 @ (suc @ I3))) => ((![I3 : nat, J2 : nat, K2 : nat]: ((ord_less_nat @ I3 @ J2) => ((ord_less_nat @ J2 @ K2) => ((P2 @ I3 @ J2) => ((P2 @ J2 @ K2) => (P2 @ I3 @ K2)))))) => (P2 @ I @ J))))))). % less_Suc_induct
thf(fact_137_strict__inc__induct, axiom,
    ((![I : nat, J : nat, P2 : nat > $o]: ((ord_less_nat @ I @ J) => ((![I3 : nat]: ((J = (suc @ I3)) => (P2 @ I3))) => ((![I3 : nat]: ((ord_less_nat @ I3 @ J) => ((P2 @ (suc @ I3)) => (P2 @ I3)))) => (P2 @ I))))))). % strict_inc_induct
thf(fact_138_not__less__less__Suc__eq, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) = (N = M)))))). % not_less_less_Suc_eq
thf(fact_139_zero__neq__one, axiom,
    ((~ ((zero_zero_poly_a = one_one_poly_a))))). % zero_neq_one
thf(fact_140_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_141_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_142_Suc__mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ (suc @ K) @ M) = (times_times_nat @ (suc @ K) @ N)) = (M = N))))). % Suc_mult_cancel1

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $true @ X @ Y) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ((q = (times_times_poly_a @ p @ zero_zero_poly_a)))).
